This question is closely related to this one: How long can language drift before it becomes indecipherable, and how to minimize drift?
Say you have a time traveler, moving forward or backward in time. The language he speaks will drift over time. Eventually, if he goes far enough forward/back in time he will find people speaking the same language, but with such significant differences from the original language as to be functionally unintelligible to him. Much as modern US citizens would have a hard time comprehending Old English.
If the time traveler didn't go quite that far. If he went far enough that language drift has caused a significant enough deviation to make communication difficult or near impossible at first, but not quite far enough to be completely unintelligible. In other words, some of the original language can still be recognized in the drifted language, even if the drift has resulted in communication problems.
My question is, how would these problems present themselves? If the traveler moved progressively further along the time-line what areas would he start struggling to understand or communicate first? Are there some concepts or methods of communication that would tend to work longer than others (ignoring gestures and body language that is universal).
If the traveler went far enough in time that he can not functionally communicate at first, would he have any advantages in deciphering the language due to the shared origin? If he were to learn to understand the new dialect over the course of a few days/weeks in this new time what tricks would he use to adopt his understanding of the language to the new dialect? or would he simply allow himself to be immersed and pick it up the way someone would if tossed in a completely foreign land?
This question is primarily about time travel, and I'm happy to accept any answers about such. However, if anyone wishes to toss in any speculation about what it would mean to have two cultures sharing a native tongue that were isolated and both languages drifted; and the difficulty of someone from one culture finally meeting a member of the other, I would welcome any feedback from that perspective as well. Specifically, would there be any extra considerations caused by having drift occurring in both culture instead of handling drift from only one culture in the time-travel scenario. However, this is only extra-goodness if you feel like contributing it; not mandatory.